We have a cute little experiment which exercises some of the equations we have discussed in this lecture. We have a mechanism by which I can make a ball perform uniform circular motion. Again for now it is not crucial how this is achieved but I can say that the string is instrumental in giving rise to the required centripetal acceleration. With a razor blade I can cut the string at which point the ball performs projectile motion with an initial velocity which matches the instantaneous velocity vector at the time the string was cut.
Performing the experiment you clearly see that without the string to induce the centripetal acceleration the ball is ejected in the tangential direction. On the table we have a carbon paper which enables us to see where the ball first hits the table. From how far it goes I can determine the magnitude of the initial velocity and I will compare that number with the number I can derive from known parameters of the uniform circular motion.
We first write an expression for the trajectory of the particle
as it is released from its circular orbit. We use Eq.
with 
(the ball comes out horizontal)
To determine 
we use our knowledge that the point of impact lies
on the trajectory. We measure that relative to the point of release
the point of impact is given by
We insert 
in Eq. 8 and solve for
to get
The other way to calculate is to use the known
parameters of the circular motion. We have
R=0.40 m and T=0.52 s which implies that
The two calculations agree well indicating that we do understand something about
projectile motion and uniform circular motion.